Breastplate and half-plate were added in late AD&D 2e to represent Renaissance-era lighter plate armor.

The summary is as follows:

• The term "breastplate" dates back to at least 1611 in the King James Bible, and was used by scholars in the 1800s to describe a piece of chest armor. It appears in AD&D as far back as 1980, and in 1995 described renaissance-era cuirass worn alone, becoming a main armor type in D&D 3.0 (2000).
• The term "full plate" dates back to at least 1909 in a book cited by Gygax himself, referring to the well-known type of full-body plate armor, and appeared in Unearthed Arcana (1985), although this type previously appeared in the original D&D boxed set as "plate armor".
• The term "half-plate" dates back to at least 1922, but did not enter D&D until 1995, where it represented a renaissance-era less bulky alternative to full plate.

Usage in D&D

Chainmail (1971) featured plate armor among its many armor types, but no breastplate or half-plate. Similarly, the original D&D boxed set's Men & Magic (1974) again featured plate armor as the strongest, but no breastplate or half-plate.

The AD&D 1e Player's Handbook (1978) introduced the term "plate mail" for its most superior armor. The Dungeon Master's Guide (1979) p.27, describes this as light chainmail with some plates, and also briefly mentions plate armor and field plate. On DMG p.165, Gygax cites Charles Ffoulkes' 1909 work Armor and Weapons as a source.

Half-plate and breastplate still do not appear in the AD&D 1e core rulebooks. As far as I can find, half-plate did not appear in any AD&D 1st edition sourcebook.

The term "breastplate" did appear in various sourcebooks, often describing a component of a more complete suit of armor. There are some instances of people wearing a breastplate alone, usually primitive tribes such as the Rovers of the Barrens (World of Greyhawk box set) or the people of Hyboria (Conan sourcebooks), and usually made of some inferior material. We do see a thief in Lankhmar sourcebook Swords of the Undercity wearing a breast plate alone.

Unearthed Arcana makes major changes to plate-type armor, and defines more clearly what each represents. Plate mail is chain or brigandine with some plates over vital areas. One point better is the new "field plate" armor, and one point better again is the "full plate" armor representing late middle ages and renaissance plates. AD&D 2e's Player's Handbook adopts these new armor types as standard.

AD&D 2e's Player's Option: Combat & Tactics (1995) introduces "half-plate" to represent a chronologically later form of plate armor, from a time when firearms became more prevalent, reducing the benefit of full-body armor. It covers the chest, outer arms, and upper legs. Historically, lower-leg armor was disadvantageous for soldiers who marched on foot, as it significantly increases the amount of energy used when marching.

Combat & Tactics also has a type of armor called "back-and-breast", which is a metal breastplate and backplate, based on armor worn during the renaissance. Historically, armor like this exists which was able to protect against a shot to the chest from early firearms, which was the most important risk to mitigate as firearms came to dominate the battlefield. Additional armor was an impractical expense and impediment to movement.

In D&D 3e's Player's Handbook, the armor list features breastplate, half-plate, and full plate. These are likely drawing on Combat & Tactics, which shares lead designer Skip Williams and was written only a few years earlier.

While D&D 4e's Player's Handbook ditched breastplate and half-plate, that edition was generally poorly received, and D&D 5e draws more from 3e. D&D 5e thus uses breastplate, half-plate and full plate. Of note, 5e recategorizes half-plate from heavy to medium armor; in 3e it was all-round inferior to full plate and never used, whereas now it's the highest-AC medium armor.

History of terminology

The word "breastplate" appears in the King James Bible, published in 1611. Revelation 9:9 says:

And they had breastplates, as it were breastplates of iron; and the sound of their wings was as the sound of chariots of many horses running to battle.

This book dated 1895 also uses it to refer to armor, suggesting (perhaps through Biblical use) it was still a word in common use at least among academics:

Sagard says that the Hurons (Iroquois) had armor made of wood. Champlain also describes the Iroquois' armor as amde of wood and thread. A plate in the same volume shows a warrior in armor. Wooden breastplates were worn. Copper breastplates have been foud, like the gold breastplates of Pero.

Breastplate explicitly referencing a metal piece of chest armor is similarly used in Andrew Lang's 1906 work Homer and His Age, an article which also quotes 1800s works on the armor worn by the Iroquois. It uses the word to describe Greek armor:

... by the seventh century B.C., a warrior could not be thought of without a breastplate; and that new poets thrust corslets and greaves into songs both new and old.

I can't find any similar historic reference to "half-plate" before 1899, so I'm skeptical that it was a term used at the time this armor was in use. However, it does pre-date D&D. Compton's Pictured Encyclopedia (1922) p.215, describes both "full plate armor" of the 16th century, and "half-plate" worn by 14th century mounted knights.

The term "full plate" appears in Charles Ffoulkes' 1909 work Armour and Weapons, which Gygax cites in the 1e DMG as a source.